J. D. Phillips, D. I. Pushkashu, A.V. Shcherbacov, Victor Shcherbacov
Abstract
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Birkhoff defined a quasigroup as an algebra (Q,⋅,\,/) that satisfies the following six identities: x⋅(x\y)=y, (y/x)⋅x=y, x\(x⋅y)=y, (y⋅x)/x=y, x/(y\x)=y, and (x/y)\x=y. We investigate triples and tetrads of identities composed of these six, emphasizing those that axiomatize the variety of quasigroups.
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Birkhoff defined a quasigroup as an algebra (Q,⋅,\,/) that satisfies the following six identities: x⋅(x\y)=y, (y/x)⋅x=y, x\(x⋅y)=y, (y⋅x)/x=y, x/(y\x)=y, and (x/y)\x=y. We investigate triples and tetrads of identities composed of these six, emphasizing those that axiomatize the variety of quasigroups.
Key concepts: Quasigroup, Mathematics, Variety (cybernetics), Axiom, Algebra over a field, Combinatorics, Pure mathematics, Discrete mathematics